Download modelling of economic cycles and maximal elements in competitive

MODELLING OF ECONOMIC CYCLES AND MAXIMAL ELEMENTS IN
COMPETITIVE ABSTRACT ECONOMIES IN THE CONTEXT OF SUSTAINABLE
DEVELOPMENT
Cristina, TĂNĂSESCU1 and Amelia, BUCUR2
Lucian Blaga University of Sibiu, e-mail: [email protected]
Lucian Blaga University of Sibiu, e-mail: [email protected]
ABSTRACT: One of the most frequently debated topics in the current specialized literature is focused on economic cycles
[1,5,6,7,8,9,10,18]. Their analysis is highly relevant for long-term predictions, in the attempt to identify solutions for economic
increase or for the understanding and solving of economic crises, in view of attaining sustainable development. Their modelling
requires the use of differential equations and periodical functions [1]. The aim of the present article is to analyse the properties of
these economic cycles, as well as historical aspects regarding mathematical modelling applied to economy. The paper will
emphasize that the modelling of economic processes resorts to statistical elements, optimization methods, differential equations, as
well as elements of the chaos theory, fixed point theory for multivoice operators; therefore these are elements of modern superior
mathematics. Let us illustrate these modellings for the situation of maximal elements in competitive abstract economies.
Keywords: economic cycles, modelling, sustainable development, competitive abstract economies, balance, demand, offer
1. INTRODUCTION
The cyclicity of the economic process can be
defined as its inherent property to evince recurrent
values of its status variables, more frequently
considered from its dynamic nature, rather than the
quantitative value. Its main qualitative determinants
are:
• alternation: the economic process subject to
cyclicity is alternately undergoing, phenomena of
economic increase of decrease;
• periodicity: the economic process subject to
cyclicity may qualitatively reverse, after an
estimated period of time, to previous values.
Mention should be made that the periodicity
(frequency) should be estimated not strictly from a
calendar perspective, but rather from an economic
perspective, over a given period of time, depending
on the evolution of certain factors of economic
cyclicity;
• inherence: the economic process shall always be
subject to cyclicity, as there is no possibility outside
this phenomenon;
• cummulativity: in the cyclical phenomenon,
alternating the economic process trend occurs on the
basis of a cummulative process. Throughout this
cummulative process, certain functionalities and/or
disfunctionalities may reach, in time, specific limits.
By reaching these limits, the economic process will
move on to the alternating trend;
• Self-adjustment: cyclicity is an economiccybernetic process characterized by self-adjustment.
Alternation of the phases of the economic cycle is a
result of the fact that once the economic process has
moved to a new phase, all the influence factors start
acting for getting out of this phase. Removal from
the phase will take place when the phase will have
reached its cumulative limits.
Thus, cyclicity represents an immanent
characteristic of the economic process, and
therefore an operating mode of this process.
2. TYPOLOGY OF THE ECONOMIC CYCLE
Specialized literature mentions several classification
criteria for the economic cycles, however, duration
is the most significant criterion. In this respect,
cycles fall into two categories: a) short-term
economic cycles (less than a year); b) long-term
economic cycles (over a year). Short-term economic
cycles are, actually, sub-annual and annual
fluctuations of the economic activity. These
fluctuations may be:
• weekly fluctuations: triggered by the weekly
alternation of some demand-related economic
variables: increased or changed expenses over the
week-end, weekly salary payment (where
applicable) etc.
• monthly fluctuations: triggered by the monthly
alternation of some demand-related economic
variables in economy: banknote circulation, various
consumption;
• seasonal fluctuations: triggered by seasonal
alternation, throughout the year, of some demandrelated variables of the economic activity:
differences in consumption (quantitative and
structural)
according
to
season,
specific
consumption behavior during holiday;
• Annual fluctuations: triggered by annual
alternations of some demand variables in economy
and they also rely on the fact that the great majority
• infrastructural Kuznets cycle (15-25 years) –
cyclical variation identified in the structure of
aggregate products may be accounted for by
prolonged underuse of the production capacities
• Kondratieff cycles: hey last for about 50 years
and represent the largest descriptive „wave” of
alternancies in the economic activities and
processes. The Kondratieff cycle (improperly
called, sometimes, secular cycle) represents a
long-term winding of short-term economic cycles
(see [18]).
Therefore, the temporary evolution of an economy
should be perceived as the result of various
fluctuations
occurring
simultaneously.
The
simultaneous existence of long-term economic
cycles, with different durations, generates the
possibility of their interaction. Hence, genuine
economic cycles are affected by a number of
alterations of their phases (either in terms of
duration or amplitude), according to their place
within a winding cycle.
Obviously, Kitchin cycles will be surrounded by
Juglar cycles which, in their turn, will be surrounded
by Kondratieff cycles (fig. 1):
of economic activities are planned and evaluated
annually. Furthermore, most institutional and
normative measure (the actual causes of economic
cyclicity) are usually taken at the beginning of the
year. These annual cycles are more pregnant in the
economies where the calendar year coincides with
the fiscal year. These fluctuations are also called
commercial cycles.
Long-term economic cycles are those cycles that last
more than a calendar year.
• Kitchin cycles: they last for about 40 months and
they are also called inventory cycles due to the
fact that the main reason for producing these
cycles is the necessity of renewing stocks of any
kind
• Juglar cycles: economic cycles proper, also called
ten-year, business or Juglar cycles, after the name
of the economist who studied them („Des crises
commerciales et leur retour periodique en France,
en Angleterre et aux Etats-Unis”, Paris, 1860).
Their duration may range from 4-5 years to 10-12
years. They are determined by the significant
evolutions of the great industrial processes
undergoing in the respective economic system, as
Kitchin
Kitchin
Juglar
Kitchin
Kitchin
Kitchin
Kitchin
Kitchin
Juglar
Kitchin
Juglar
Kitchin
Kitchin
Kitchin
Juglar
Kitchin
Kitchin
Kitchin
Juglar
Kondratieff
well as the long-term banking policies.
As we have already mentioned above, this winding
has an impact on the shape, amplitude and duration
of the cycles phases that are being wound in longer
cycles. More specifically, a cycle wound throughout
the expansion phase of an winding cycle will evince
longer duration and greater amplitude of its
expansion phase and, respectively, shorter duration
and smaller amplitude in its recession phase.
Reversely, a cycle that is being wound in the
recession phase of an winding cycle will have an
expansion phase with shorter duration and smaller
amplitude and, respectively, a recession phase with
longer duration and greater amplitude. This is a
highly significant result for drawing up long-term
policies, especially regarding the restructuring
processes or deep economic changes, since it may
use the historical „location” where a particular
economic system and process are situated, in view of
capitalizing specific advantages. Should we resort to
a metaphor in this matter we might say that
economic systems have to use economic „currents”,
just like birds do, in order to maximize their effort
and accomplish their target.
3. INTEGRATING BUSINESS CYCLES INTO
ECONOMIC CYCLES
Business cycles are called endogenous or
deterministic, whenever they are generated by the
internal mechanism of economy. The term is usually
employed to make a clear distinction from
traditional models, where fluctuations are triggered
by exogenous, most often, random shocks.
There is a great number of studies on empirical
methods of testing mechanisms generating nonlinear data. Certainly, the theory of business cycles
and empirical methods has recorded a long tradition
by now. Cycle persistence has been identified by
economists as early as the 19th century, whereas
thorough theories of fluctuations and business cycles
have started to emerge in the fourth and fifth
decades of the 20th century, due to the contributions
of Frisch (1933), Kalecki (1937), Samuelson (1939),
Kaldor (1940), Hicks (1950) and Goodwin (1951).
As early as that time, two potential perspectives
regarding the business cycles analysis have
emerged: their creation either by oscillation-
generating mechanisms (the so-called non-linear
endogenous cycles) or by means of the randomshock impact on a fundamentally stable economic
system (the so-called Slutsky-Frisch stochastic
cycles).
Before 1970s, the great majority of econometrists
focused on the Slutsky-Frisch-type tradition of the
business stochastic cycles by means of linead
econometric
techniques.
They
overlooked
endogenous cycles and rarely focused on the nonlinear data-generating mechanism (except for the
studies undertaken by Klein and Preston, 1969).
Studies on non-linearities have rapidly developed
recently. It focused on the analysis of univarious and
multivarious business cycles, and the methods
attempted
to
identify the data-generating
mechanisms for oscillating or chaotic movements
(Blatt, 1978, Brock, 1986, şi Ramsey, 1988).
The new statistical methods helped discover
significant non-linear structures in the data series in
the fields of economy and finances (Tong, 1990;
Brock, Hsie şi LeBaron, 1991; Brock, 1992; Granger
şi Terasvirta, 1993) as well as significant non-linear
structure have been modelled. Mention should be
made, in Romanian specialized studies, of the
studies authored by academician Tudorel Postolache
(1981), as early as the 1980s, on the typology and
characteristics of cycles approached from a
historical perspective.
The first systematic approach to the chronological
series regarding business cycles was accomplished
by Burns and Mitchell (1946). Most recent studies,
however, started from the premise that variables
operate according to linear stochastic processes with
consistent coefficients. Such an approach allows a
better integration of the macroeconomic theory with
econometrics. In spite of this integration and a
proper understanding of statistical properties,
compared to the „richness” of Burns-Mitchell
analysis, the potential analysis of asymmetries
between recessions and expansions or the temporal
notion of business cycles (as opposed to calendar
cycles) have disappeared.
The major problem of macroeconomists as regards
the fluctuation description resided in the distinction
of the trend from the cyclic component. Although
this separation may be interpreted from a purely
statistical perspective, many economists still believe
that, given the absence of short-term fluctuations,
the economy will evince an overall increasing trend
which might be considered as the one aimed at.
Economic historians have also identified some
distinct cycles such as the short-term business cycle
(3-7 years), the construction or Kuznets cycle (15-2
years), as well as the long-wave of Kondratieff cycle
(40-60 years). There is the natural question,
therefore, how can different cycles interact?
First and foremost, economic systems differ from
most other systems in the natural sciences due to the
prevailing character of positive reverse reaction
loops. Best known examples in this respect refer to
the accelerator and the Keynes theory of multiplying
loops.
Other positive loops operate in the case of the socalled extrapolative expectations, crowding-out
effect, increased efficiency, the effect of inflationist
expectations on the real interest rates and therefore
the aggregate demands, the financial crises and
speculations as well as the synergy among and
within the technologies employed for production,
communications and organizations (Graham and
Senge, 1980, Sterman, 1986, Arthur, 1988,
Semmler, 1989, a.s.o.).
It is obvious thus that the development of non-linear
concepts and methods represents a product of the
computer era. Most studies in the initial stages
started from the numerical analysis of extremely
simple non-linear models, what nowadays are
merely the fundamentals of data processing and PC
calculation. The conclusion reached was that even
the simplest non-linear models are able to reproduce
a wide variety of characteristics.
For instance, one conclusion was that small changes
in the parameter values lead to surprising results,
such as simple classical models, previously
considered as having a cyclical and even easily
predictible behaviour. This has entailed the idea of
„determinist chaos”. Another significant feature of
logistics, whenever employing its representation as
bifurcation diagram, is that every time a window is
closely looked at (higher resolution), the diagram is
represented as an identical copy of the integral
diagram.
The analysis highlighted the great diversity of
characteristics of an apparently simple model, where
the standard idea of a static balance, intensively
employed in linear economic models, can describe
only the very limited area of possibilities of a
monotomous behaviour or with regular oscillations.
For the past few years, the discoveries regarding the
theory of non-linear dynamic systems have
generated significant changes in the classical theory
of economics. The for non-linear and deterministic
systems has continuously intensified and related
studies have developed.
There are at least two reasons that make chaos
theory so appealing in order to explain the economic
phenomenon. The former is the interest evinced by
scientists for creating realistic models. Most
characteristic of the chaos theory are obviously in
contradiction with fundamental notions of classical
economy, the very foundation of contemporary
economic theory.
The classical theory defines the universe as a
mechanism that might be subject to accurate
measurements, precise predictions and control.
Chaos, on the other hand, is a deterministic system
characterized by complex behaviour, based on
apparently hazardous interaction among elements.
The latter is the possibility of controlling systems.
The initial opinion upheld by the scientists was that
the chaotic movement of systems is neither
predictable nor controllable because of the sensitive
dependence to initial circumstances. Small changes
lead to other chaotic movements, not to any stable or
predictable alternative.
Scientists managed to demonstrate, however, that
chaos may be unpredictable but its deterministic
structure makes it exploitable and consequently
controllable. Paradoxically, the sensitivity of the
chaotic system to initial circumstances may be used
to control it. This observation has opened up the
possibility of changing the behavior of natural
systems without altering any of the initial
characteristics. Nevertheless, chaos theory has
prompted economists to study the following
contradictions to classical theory: economics is a
non-linear science, its dynamics leads to multiple
solutions, and economic units are nonrepresentatives. Compared to the models of neoclassical theory, the application of chaos theory to
economics enables researchers to interpret
phenomena previously considered stochastic,
exogenous and unlikely to be influenced [5,6,7,8].
4. COMPETITIVE ABSTRACT ECONOMIES
The aim of this section is to highlight some of the
most significant economic models. Let us mention
some mathematical results, especially those related
to the analysis of multivocal operators, since the
results have particular relevance in the context of
various economic problems.
The Arrow-Debreu model (1959) has been designed
for an economy considered to have a finite number
of companies. This is one of the most significant
studies in the modern theory of general balance and
it refers to a Walrasian general balance. The
economic interpretation of the Walras law is that in a
closed economy almost the entire individual income
will be spent. There are two illustrations of the
Walrasian balance in specialized studies. The former
uses the classical notion of request and offer
functions. A price vector is considered to be
balanced provided that the request equals the offer.
The latter is independent from the request and offer
functions.
In the classical Arrow-Debreu model, a finite
number of goods are traded, produced or consumed.
As regards merchandise such as: steel, cereal or
apples, these are available for various time intervals,
locations or states. Let us consider there are l such
merchandise. Then it is normal to look at the E area
of merchandise as a sub-multitude of . The
consumers` preference more instead of less has
significant consequences for the balance analysis.
On the other hand, the fundamental principle of
theoretical economy is that agents behave rationally,
i.e. they are aware of their own interest and their
actions are aimed at maximizing profit [2,11].
An x vector from E includes on each component the
product quantity that is subject to a specific
transaction at a given time. Therefore, it is not
individual products but rather these vectormerchandise that will be produced, traded or
consumed. Of course, some of the x components
may be null.
Let p stand for the price vector, i.e. the vector that
includes on each component the value of a product
unit. The value of x may then be calculated as a
scalar product as follows:
, = ∑
(1)
The working hypothesis is a finite number of
consumers.
Certainly, not every merchandise vector is available
as final consumption for a consumer. Therefore
⊆ stands for the multitude of all merchandise
vector available to the i consumer, i.e. its
consumption multitude. Let us remember that the
admissible merchandise vector are positive.
Let us consider the budget multitude of the i
consumer, i.e. the multitude of those merchandise
vectors of the consumption multitude available and
affordable at a p price and budget.
= |, ≤ (2)
Certainly the budget multitude may also be void. In
the context of a market economy, any consumer is
confronted with the problem of selecting one or
more elements from the budget multitude. The
choice of one or another of the elements of this
multitude shall be made according to a given
criterion. One method of defining such a criterion
requires the use and existence of a utility function
: → . The i consumer will choose the x over y
merchandise vector provided that > and
it makes no difference which one to choose if
= . Therefore, the mathematical
problem of the consumer is to maximize the utility
function on the budget multitude.
Mathematical analysis shows that if the utility
function is continuous and the multitude is
compact, on the basis of the Weierstrass theorem,
then there are x vectors that maximize the function on the multitude. However, economic
practice shows that such constrains are too harsh and
there are lighter hypotheses of consumer preferences
that enable the problem solution. The description of i
consumer preference will be accomplished by means
of the multivocal operator of preferences : →
. For instance,
= |
> = | este strict preferat
will be selected. It is obvious that ∗ ∈ is the
optimum preference if ∗ is void. Such an
element is called U – maximal. One can easily
demonstrate that any fixed point theorem for
multivocal operator generates an existence result of
a U - maximal point for the preference function
[13,15,16].
5. SUSTAINABLE DEVELOPMENT BY
EFFECTIVE ECONOMIC ACTIVITIES.
CONCLUSIONS
One of the methods of sustainable development is to
make economic activities more effective,
considering the urgency of monimizing production
costs, thus entailing a complex capitalization of raw
materials, minimizing the consumption raw
materials well as oil and energy [12].
The market economy is performative by excellence,
due to the fact that it is free and competitive. Given
the lack of competitiveness, the product quality will
diminish accompanied, paradoxically, by an increase
in the price. Thus the „amazing” result is poor
quality products available on the market at excessive
prices, associated with very good „sales” – because
of their low reliability, products (parts) have to be
replaced very often in the production processes.
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